Simplify and expand the following expression: $ \dfrac{5z - 2}{2z + 2}-\dfrac{z}{3z - 5} $
Solution: In order to subtract expressions, they must have a common denominator. Get both fractions over a common denominator of $(2z + 2)(3z - 5)$ Multiply the first term by $\dfrac{3z - 5}{3z - 5}$ $ \begin{align*} \dfrac{5z - 2}{2z + 2} \times \dfrac{3z - 5}{3z - 5} & = \dfrac{(5z - 2)(3z - 5)}{(2z + 2)(3z - 5)} \\ & = \dfrac{15z^2 - 31z + 10}{(2z + 2)(3z - 5)}\end{align*} $ Multiply the second term by $\dfrac{2z + 2}{2z + 2}$ $ \begin{align*} \dfrac{z}{3z - 5} \times \dfrac{2z + 2}{2z + 2} & = \dfrac{(z)(2z + 2)}{(3z - 5)(2z + 2)} \\ & = \dfrac{2z^2 + 2z}{(3z - 5)(2z + 2)}\end{align*} $ Now we have: $ = \dfrac{15z^2 - 31z + 10}{(2z + 2)(3z - 5)} - \dfrac{2z^2 + 2z}{(3z - 5)(2z + 2)} $ Now both terms have a common denominator we can subtract the numerators: $ = \dfrac{15z^2 - 31z + 10 - (2z^2 + 2z)}{(2z + 2)(3z - 5)} $ $ = \dfrac{15z^2 - 31z + 10 - 2z^2 - 2z}{(2z + 2)(3z - 5)} $ $ = \dfrac{13z^2 - 33z + 10}{(2z + 2)(3z - 5)}$ Expand the denominator: $ = \dfrac{13z^2 - 33z + 10}{6z^2 - 4z - 10}$